We introduce a new regression framework designed to deal with large-scale, complex data that lies around a low-dimensional manifold. Our approach first constructs a graph representation, referred to as the skeleton, to capture the underlying geometric structure. We then define metrics on the skeleton graph and apply nonparametric regression techniques, along with feature transformations based on the graph, to estimate the regression function. In addition to the included nonparametric methods, we also discuss the limitations of some nonparametric regressors with respect to the general metric space such as the skeleton graph. The proposed regression framework allows us to bypass the curse of dimensionality and provides additional advantages that it can handle the union of multiple manifolds and is robust to additive noise and noisy observations. We provide statistical guarantees for the proposed method and demonstrate its effectiveness through simulations and real data examples.

翻译：我们提出了一种新的回归框架，旨在处理分布于低维流形周围的大规模复杂数据。该方法首先构建一个称为“骨架”的图表示，以捕捉潜在的几何结构。随后，我们在骨架图上定义度量，并应用非参数回归技术以及基于图的特征变换来估计回归函数。除所包含的非参数方法外，我们还讨论了某些非参数回归器在通用度量空间（如骨架图）中的局限性。所提出的回归框架能够规避维度灾难，并具备额外优势：可处理多个流形的并集，且对加性噪声和含噪观测具有鲁棒性。我们为该方法的统计保证提供了理论依据，并通过模拟实验和真实数据示例验证了其有效性。